Baire space

<h2 id="definition">Definition</h2>
The definition that follows is based on the notions of <a href="/facts/Meagre_set/AxTk8frw">meagre</a> (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and <a href="/facts/Nonmeagre/AxTk8frw">nonmeagre</a> (or second category) set (namely, a set that is not meagre). See the corresponding article for details.
A topological space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is called a Baire space if it satisfies any of the following equivalent conditions:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a><a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<ol><li>Every countable intersection of <a href="/facts/Dense_(topology)/nPT9Yxao">dense</a> <a href="/facts/Open_set/QfTCIqMu">open sets</a> is dense.</li>
<li>Every countable union of closed sets with empty interior has empty interior.</li>
<li>Every meagre set has empty interior.</li>
<li>Every nonempty open set is nonmeagre.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>Every <a href="/facts/Comeagre/AxTk8frw">comeagre</a> set is dense.</li>
<li>Whenever a countable union of closed sets has an interior point, at least one of the closed sets has an interior point.</li></ol>
The equivalence between these definitions is based on the associated properties of complementary subsets of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 (that is, of a set 
 
 
 
 A
 ⊆
 X
 
 
 {\displaystyle A\subseteq X}
 
 and of its <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a> 
 
 
 
 X
 ∖
 A
 
 
 {\displaystyle X\setminus A}
 
) as given in the table below.

<table><tbody><tr><th>Property of a set</th><th>Property of complement</th></tr><tr><td>open</td><td>closed</td></tr><tr><td>comeagre</td><td>meagre</td></tr><tr><td>dense</td><td>has empty interior</td></tr><tr><td>has dense interior</td><td>nowhere dense</td></tr></tbody></table>
<h2 id="baire-category-theorem">Baire category theorem</h2>
Main article: <a href="/facts/Baire_category_theorem/AwodvqgV">Baire category theorem</a>
The Baire category theorem gives sufficient conditions for a topological space to be a Baire space.

<ul><li>(BCT1) Every <a href="/facts/Complete_metric_space/lsckmU8G">complete</a> <a href="/facts/Pseudometric_space/uy1o2lhk">pseudometric space</a> is a Baire space.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> In particular, every <a href="/facts/Completely_metrizable/JbJaIzI8">completely metrizable</a> topological space is a Baire space.</li>
<li>(BCT2) Every <a href="/facts/Locally_compact_regular/hesalT7q">locally compact regular</a> space is a Baire space.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> In particular, every <a href="/facts/Locally_compact_Hausdorff/hesalT7q">locally compact Hausdorff</a> space is a Baire space.</li></ul>
BCT1 shows that the following are Baire spaces:

<ul><li>The space 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 of <a href="/facts/Real_number/R02gw5Pb">real numbers</a>.</li>
<li>The space of <a href="/facts/Irrational_number/Psv19TET">irrational numbers</a>, which is <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to the <a href="/facts/Baire_space_(set_theory)/WPtHVtTP">Baire space 
 
 
 
 
 ω
 
 ω
 
 
 
 
 {\displaystyle \omega ^{\omega }}
 
 of set theory</a>.</li>
<li>Every <a href="/facts/Polish_space/CkUFcGap">Polish space</a>.</li></ul>
BCT2 shows that the following are Baire spaces:

<ul><li>Every compact Hausdorff space; for example, the <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a> (or <a href="/facts/Cantor_space/h1sNqjBT">Cantor space</a>).</li>
<li>Every <a href="/facts/Manifold/rXPERJtu">manifold</a>, even if it is not <a href="/facts/Paracompact/rjTrvF7j">paracompact</a> (hence not <a href="/facts/Metrizable/BCLdo5Jf">metrizable</a>), like the <a href="/facts/Long_line_(topology)/bgkMDrpB">long line</a>.</li></ul>
One should note however that there are plenty of spaces that are Baire spaces without satisfying the conditions of the Baire category theorem, as shown in the Examples section below.

<h2 id="properties">Properties</h2>
<ul><li>Every nonempty Baire space is nonmeagre. In terms of countable intersections of dense open sets, being a Baire space is equivalent to such intersections being dense, while being a nonmeagre space is equivalent to the weaker condition that such intersections are nonempty.</li>
<li>Every open subspace of a Baire space is a Baire space.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li>
<li>Every dense <a href="/facts/G-delta_set/Erg1BCFk">Gδ set</a> in a Baire space is a Baire space.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> The result need not hold if the Gδ set is not dense. See the Examples section.</li>
<li>Every comeagre set in a Baire space is a Baire space.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a></li>
<li>A subset of a Baire space is comeagre if and only if it contains a dense Gδ set.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a></li>
<li>A closed subspace of a Baire space need not be Baire. See the Examples section.</li>
<li>If a space contains a dense subspace that is Baire, it is also a Baire space.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a></li>
<li>A space that is locally Baire, in the sense that each point has a neighborhood that is a Baire space, is a Baire space.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a><a class="footnote-ref" id="fnref:22" href="#fn:22">22</a></li>
<li>Every <a href="/facts/Topological_sum/4SLSRbLt">topological sum</a> of Baire spaces is Baire.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a></li>
<li>The product of two Baire spaces is not necessarily Baire.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a></li>
<li>An arbitrary product of complete metric spaces is Baire.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a></li>
<li>Every <a href="/facts/Locally_compact/hesalT7q">locally compact</a> <a href="/facts/Sober_space/Ffmz3RlE">sober space</a> is a Baire space.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a></li>
<li>Every finite topological space is a Baire space (because a finite space has only finitely many open sets and the intersection of two open dense sets is an open dense set<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>).</li>
<li>A <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> is a Baire space if and only if it is nonmeagre,<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> which happens if and only if every closed balanced absorbing subset has non-empty interior.<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a></li></ul>
Let 
 
 
 
 
 f
 
 n
 
 
 :
 X
 →
 Y
 
 
 {\displaystyle f_{n}:X\to Y}
 
 be a sequence of <a href="/facts/Continuous_map_(topology)/sKbl02pB">continuous</a> functions with pointwise limit 
 
 
 
 f
 :
 X
 →
 Y
 .
 
 
 {\displaystyle f:X\to Y.}
 
 If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a Baire space, then the points where 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is not continuous is a <a href="/facts/Meagre_set/AxTk8frw">meagre set</a> in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and the set of points where 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is continuous is dense in 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
 A special case of this is the <a href="/facts/Uniform_boundedness_principle/rbbdebsr">uniform boundedness principle</a>.

<h2 id="examples">Examples</h2>
<ul><li>The empty space is a Baire space. It is the only space that is both Baire and meagre.</li>
<li>The space 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> with the usual topology is a Baire space.</li>
<li>The space 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> (with the topology induced from 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
) is not a Baire space, since it is meagre.</li>
<li>The space of <a href="/facts/Irrational_number/Psv19TET">irrational numbers</a> (with the topology induced from 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
) is a Baire space, since it is comeagre in 
 
 
 
 
 R
 
 .
 
 
 {\displaystyle \mathbb {R} .}
 
</li>
<li>The space 
 
 
 
 X
 =
 [
 0
 ,
 1
 ]
 ∪
 (
 [
 2
 ,
 3
 ]
 ∩
 
 Q
 
 )
 
 
 {\displaystyle X=[0,1]\cup ([2,3]\cap \mathbb {Q} )}
 
 (with the topology induced from 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
) is nonmeagre, but not Baire. There are several ways to see it is not Baire: for example because the subset 
 
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle [0,1]}
 
 is comeagre but not dense; or because the nonempty subset 
 
 
 
 [
 2
 ,
 3
 ]
 ∩
 
 Q
 
 
 
 {\displaystyle [2,3]\cap \mathbb {Q} }
 
 is open and meagre.</li>
<li>Similarly, the space 
 
 
 
 X
 =
 {
 1
 }
 ∪
 (
 [
 2
 ,
 3
 ]
 ∩
 
 Q
 
 )
 
 
 {\displaystyle X=\{1\}\cup ([2,3]\cap \mathbb {Q} )}
 
 is not Baire. It is nonmeagre since 
 
 
 
 1
 
 
 {\displaystyle 1}
 
 is an isolated point.</li></ul>
The following are examples of Baire spaces for which the Baire category theorem does not apply, because these spaces are not locally compact and not completely metrizable:

<ul><li>The <a href="/facts/Sorgenfrey_line/uE1nTKxG">Sorgenfrey line</a>.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a></li>
<li>The <a href="/facts/Sorgenfrey_plane/Q3vWuzgx">Sorgenfrey plane</a>.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a></li>
<li>The <a href="/facts/Niemytzki_plane/vNMfxaBY">Niemytzki plane</a>.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a></li>
<li>The subspace of 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
 consisting of the open upper half plane together with the rationals on the x-axis, namely, 
 
 
 
 X
 =
 (
 
 R
 
 ×
 (
 0
 ,
 ∞
 )
 )
 ∪
 (
 
 Q
 
 ×
 {
 0
 }
 )
 ,
 
 
 {\displaystyle X=(\mathbb {R} \times (0,\infty ))\cup (\mathbb {Q} \times \{0\}),}
 
 is a Baire space,<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a> because the open upper half plane is dense in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and completely metrizable, hence Baire. The space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is not locally compact and not completely metrizable. The set 
 
 
 
 
 Q
 
 ×
 {
 0
 }
 
 
 {\displaystyle \mathbb {Q} \times \{0\}}
 
 is closed in 
 
 
 
 X
 
 
 {\displaystyle X}
 
, but is not a Baire space. Since in a metric space closed sets are <a href="/facts/G-delta_set/Erg1BCFk">Gδ sets</a>, this also shows that in general Gδ sets in a Baire space need not be Baire.</li></ul>
<a href="/facts/Algebraic_varieties/Vk7f97vn">Algebraic varieties</a> with the <a href="/facts/Zariski_topology/uolkKBpM">Zariski topology</a> are Baire spaces. An example is the affine space 
 
 
 
 
 
 A
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {A} ^{n}}
 
 consisting of the set 
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {C} ^{n}}
 
 of n-tuples of complex numbers, together with the topology whose closed sets are the vanishing sets of polynomials 
 
 
 
 f
 ∈
 
 C
 
 [
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ]
 .
 
 
 {\displaystyle f\in \mathbb {C} [x_{1},\ldots ,x_{n}].}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Banach%25E2%2580%2593Mazur_game/FImYgJat">Banach–Mazur game</a></li>
<li><a href="/facts/Barrelled_space/r2H58TN3">Barrelled space</a> – Type of topological vector space</li>
<li><a href="/facts/Blumberg_theorem/h015Form">Blumberg theorem</a> – Any real function on R admits a continuous restriction on a dense subset of R</li>
<li><a href="/facts/Choquet_game/TD1VYUvY">Choquet game</a> – Topological game</li>
<li><a href="/facts/Property_of_Baire/G2Sk5Azz">Property of Baire</a> – Difference of an open set by a meager set</li>
<li><a href="/facts/Webbed_space/ChtDnm0X">Webbed space</a> – Space where open mapping and closed graph theorems hold</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, Nicolas</a> (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. <a href="/facts/%25C3%2589l%25C3%25A9ments_de_math%25C3%25A9matique/8TQroWxE">Éléments de mathématique</a>. Vol. 4. Berlin New York: Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-64563-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/246032063">246032063</a>.</li>
<li><a href="/facts/Ryszard_Engelking/aAjuMWvC">Engelking, Ryszard</a> (1989). General Topology. Heldermann Verlag, Berlin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-88538-006-4.</li>
<li>Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; <a href="/facts/Dana_Scott/XEVddlVg">Scott, D. S.</a> (2003). <a href="https://archive.org/details/continuouslattic0000unse">Continuous Lattices and Domains</a>. Encyclopedia of Mathematics and its Applications. Vol. 93. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0521803380.</li>
<li>Haworth, R. C.; McCoy, R. A. (1977), <a href="http://eudml.org/doc/268479">Baire Spaces</a>, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk</li>
<li><a href="/facts/John_L._Kelley/zmQyF2xn">Kelley, John L.</a> (1975) [1955]. General Topology. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 27 (2nd ed.). New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90125-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/1365153">1365153</a>.</li>
<li><a href="/facts/James_Munkres/Ybfa6fs4">Munkres, James R.</a> (2000). Topology. <a href="/facts/Prentice_Hall/bynWAuat">Prentice-Hall</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-13-181629-2.</li>
<li>Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li>
<li><a href="/facts/Eric_Schechter/cApyLyxB">Schechter, Eric</a> (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-622760-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/175294365">175294365</a>.</li>
<li><a href="/facts/Albert_Wilansky/I6QRVtGa">Wilansky, Albert</a> (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-49353-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/849801114">849801114</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.encyclopediaofmath.org/index.php/Baire_space">Encyclopaedia of Mathematics article on Baire space</a></li>
<li><a href="http://www.encyclopediaofmath.org/index.php/Baire_theorem">Encyclopaedia of Mathematics article on Baire theorem</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Munkres 2000, p. 295. - Munkres, James R. (2000). Topology. Prentice-Hall. ISBN 0-13-181629-2. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Your favourite application of the Baire Category Theorem". Mathematics Stack Exchange. <a href="https://math.stackexchange.com/q/165696" target="_blank">https://math.stackexchange.com/q/165696</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Classic applications of Baire category theorem". MathOverflow. <a href="https://mathoverflow.net/questions/129666" target="_blank">https://mathoverflow.net/questions/129666</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Engelking 1989, Historical notes, p. 199. - Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Bourbaki 1989, p. 192. - Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. Vol. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063. <a href="https://search.worldcat.org/oclc/246032063" target="_blank">https://search.worldcat.org/oclc/246032063</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Baire, R. (1899). "Sur les fonctions de variables réelles". Annali di Matematica Pura ed Applicata. 3: 1–123. doi:10.1007/BF02419243. <a href="https://books.google.com/books?id=cS4LAAAAYAAJ" target="_blank">https://books.google.com/books?id=cS4LAAAAYAAJ</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Munkres 2000, p. 295. - Munkres, James R. (2000). Topology. Prentice-Hall. ISBN 0-13-181629-2. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Haworth & McCoy 1977, p. 11. - Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk <a href="http://eudml.org/doc/268479" target="_blank">http://eudml.org/doc/268479</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Narici & Beckenstein 2011, pp. 390–391. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">As explained in the meagre set article, for an open set, being nonmeagre in the whole space is equivalent to being nonmeagre in itself. <a href="/wiki/Meagre_set" target="_blank">/wiki/Meagre_set</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Kelley 1975, Theorem 34, p. 200. - Kelley, John L. (1975) [1955]. General Topology. Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90125-1. OCLC 1365153. <a href="https://search.worldcat.org/oclc/1365153" target="_blank">https://search.worldcat.org/oclc/1365153</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Schechter 1996, Theorem 20.16, p. 537. - Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. <a href="https://search.worldcat.org/oclc/175294365" target="_blank">https://search.worldcat.org/oclc/175294365</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Kelley 1975, Theorem 34, p. 200. - Kelley, John L. (1975) [1955]. General Topology. Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90125-1. OCLC 1365153. <a href="https://search.worldcat.org/oclc/1365153" target="_blank">https://search.worldcat.org/oclc/1365153</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Schechter 1996, Theorem 20.18, p. 538. - Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. <a href="https://search.worldcat.org/oclc/175294365" target="_blank">https://search.worldcat.org/oclc/175294365</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Haworth & McCoy 1977, Proposition 1.14. - Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk <a href="http://eudml.org/doc/268479" target="_blank">http://eudml.org/doc/268479</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Haworth & McCoy 1977, Proposition 1.23. - Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk <a href="http://eudml.org/doc/268479" target="_blank">http://eudml.org/doc/268479</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Ma, Dan (3 June 2012). "A Question About The Rational Numbers". Dan Ma's Topology Blog.Theorem 3 <a href="https://dantopology.wordpress.com/2012/06/02/a-question-about-the-rational-numbers/" target="_blank">https://dantopology.wordpress.com/2012/06/02/a-question-about-the-rational-numbers/</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Haworth & McCoy 1977, Proposition 1.16. - Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk <a href="http://eudml.org/doc/268479" target="_blank">http://eudml.org/doc/268479</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Haworth & McCoy 1977, Proposition 1.17. - Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk <a href="http://eudml.org/doc/268479" target="_blank">http://eudml.org/doc/268479</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Haworth & McCoy 1977, Theorem 1.15. - Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk <a href="http://eudml.org/doc/268479" target="_blank">http://eudml.org/doc/268479</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Narici & Beckenstein 2011, Theorem 11.6.7, p. 391. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Haworth & McCoy 1977, Corollary 1.22. - Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk <a href="http://eudml.org/doc/268479" target="_blank">http://eudml.org/doc/268479</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Haworth & McCoy 1977, Proposition 1.20. - Haworth, R. C.; McCoy, R. A. (1977), Baire Spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk <a href="http://eudml.org/doc/268479" target="_blank">http://eudml.org/doc/268479</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Oxtoby, J. (1961). "Cartesian products of Baire spaces" (PDF). Fundamenta Mathematicae. 49 (2): 157–166. doi:10.4064/fm-49-2-157-166. <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm49/fm49113.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/fm/fm49/fm49113.pdf</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Fleissner, W.; Kunen, K. (1978). "Barely Baire spaces" (PDF). Fundamenta Mathematicae. 101 (3): 229–240. doi:10.4064/fm-101-3-229-240. <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm101/fm101121.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/fm/fm101/fm101121.pdf</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Bourbaki 1989, Exercise 17, p. 254. - Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. Vol. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063. <a href="https://search.worldcat.org/oclc/246032063" target="_blank">https://search.worldcat.org/oclc/246032063</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Gierz et al. 2003, Corollary I-3.40.9, p. 114. - Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S. (2003). Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications. Vol. 93. Cambridge University Press. ISBN 978-0521803380. <a href="https://archive.org/details/continuouslattic0000unse" target="_blank">https://archive.org/details/continuouslattic0000unse</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">"Intersection of two open dense sets is dense". Mathematics Stack Exchange. <a href="https://math.stackexchange.com/q/1143211" target="_blank">https://math.stackexchange.com/q/1143211</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Narici & Beckenstein 2011, Theorem 11.8.6, p. 396. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. <a href="https://search.worldcat.org/oclc/144216834" target="_blank">https://search.worldcat.org/oclc/144216834</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Wilansky 2013, p. 60. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. <a href="https://search.worldcat.org/oclc/849801114" target="_blank">https://search.worldcat.org/oclc/849801114</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">"The Sorgenfrey line is a Baire Space". Mathematics Stack Exchange. <a href="https://math.stackexchange.com/q/476821" target="_blank">https://math.stackexchange.com/q/476821</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">"The Sorgenfrey plane and the Niemytzki plane are Baire spaces". Mathematics Stack Exchange. <a href="https://math.stackexchange.com/q/3848442" target="_blank">https://math.stackexchange.com/q/3848442</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">"The Sorgenfrey plane and the Niemytzki plane are Baire spaces". Mathematics Stack Exchange. <a href="https://math.stackexchange.com/q/3848442" target="_blank">https://math.stackexchange.com/q/3848442</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">"Example of a Baire metric space which is not completely metrizable". Mathematics Stack Exchange. <a href="https://math.stackexchange.com/q/3003649" target="_blank">https://math.stackexchange.com/q/3003649</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
</ol>

Baire space open-in-new

Baire space